## Dynamical Systems

### September 5, 2009

Dynamical systems are mathematical models of real-world phenomena, such as weather or the oceans, which change with time.

The difference map of a dynamical system specifies how one state of the model changes into the next state.

If the system changes continuously, its map may be defined with differential equations. If the system changes from one state to another in steps, it is called a discrete dynamical system, and its map may be described with difference equations. This discussion will concentrate exclusively on discrete systems.

So a map is a function f which takes the current state of the system x to its next state, x’. The domain of the map is thus all possible system states. This set of all possible system states is called phase space.